Discovery and Invention Part I: Distinctions and Notations

In this three part (probably) series, I’m going to look at the notions of invention and discovery as they relate to how we think about mathematics and logic. In this first post, I’m going to set up the distinction between discovery and invention as I see it, and then talk about whether systems of notation for logic and math are discovered or invented. The second part will look at how our philosophical views affect whether we think math is invented or discovered, and the third will do something similar for logic. (links will be added as the posts are finished)

Discovery vs. Invention

In science (especially), there are some sorts of things that are clearly discoveries: “We discovered this weird fish at the bottom of the arctic ocean”, “We discovered a molecule that kills cancer cells”, but “Someone invented an artificial heart”; “Someone invented a more efficient fuel injector”.

It seems odd to say you invented a molecule unless you really did construct it for the first time in a lab, and it sounds odd to say someone discovered a mechanical device if they were the first to construct or design that device.

What it comes down to, broadly speaking, is whether something was out there already or whether it’s been created for the first time (or at least independently of similar inventions). What first comes to mind when someone mentions inventions are probably (types of – I’ll suppress this qualifier from now on) physical objects — space shuttles, the wheel — or materials — plastics, alloys. But we also sometimes talk of people inventing characters, musical themes, or abstract systems (government structures for example).

In all of those cases I would class the objects of invention as artifacts. In this context I will take a artifact to be an abstract or physical object (or type of object) that is dependent on minds (human, animal, or alien) for its initial existence. (Depending on exactly how we want to cash out “invention” this definition may turn out to be circular, but it good enough for what I want to say here.)

The objects of discovery, on the other hand, are mind independent — they would have been there whether or not we were around to find them. Planets, quarks and fish are the sorts of things that can be discovered but not invented.

That said, we also speak of archaeological discoveries and discoveries of maps, or even previous inventions. In all of these cases what’s being discovered is something created by (i.e. dependent on), human minds, albeit minds far removed from our own. This suggests a broader reading of meaning of ‘discover’ along the lines of ‘find something previously unknown (to the discoverer)’. So then we might say that to discover something is to find something not known for a significant time (if ever). In any case, math and logic are unlikely to fall into the category of things created by temporally and cognitively distant minds and found again in modern times, so I’ll spend no more time on analysis here.

There are two further assumptions I’m going to make about invention and discovery. Either could be denied, but I find them plausible. First, I take invention and discovery to be mutually exclusive in the following sense: one can’t be the discoverer and inventor of the same object. Again, a previous invention can be discovered, but the original object is an invention not a discovery.

Second, I don’t take invention and discovery to be exhaustive. There are things we create that aren’t inventions — new thoughts, new combinations of words or Lego, works of art. . . It would also, I think, be odd to think of people discovering trees or rain — ubiquitous natural occurrences don’t seem like the sort of thing we, as people on Earth, could ‘discover’.

Systems of Notation

Now that we have some sort of handle on the difference between discovery and invention, the first thing I want to get out of the way is the fairly straightforward case of notation systems. It is clear to me that these are invented. In the case of logic, we can (and do) represent the same content in Frege notation, Polish notation and standard Peano-Russell notation. Even within the latter we might represent conjunction as or as without loss or change of meaning.

The case of mathematics is similar. When doing calculus, we’re generally happy to move back and forth between Newton’s notation () and Leibniz’s notation () based on convenience. Likewise, we can express numbers using Roman or western Arabic numerals depending on what century we’re living in.

It is also clear that had Frege never been born, Frege notation (probably) wouldn’t have existed, which is to say that the existence of particular notation systems is dependent on minds (though generally not one particular mind).

One way to think about mind-independence is using couterfactuals: if there hadn’t been human’s, would there be logical notation systems? The answer, surely, is ‘no’.

It should be clear then, that notation systems aren’t discovered. Are they invented? We generally think of inventions as new creations, perhaps requiring some certain level of novelty and/or usefulness. If you’ve ever seen Frege notation, you have to agree that it’s novel, and if you’ve ever used the calculus, you’d probably find that having some system of notation is indispensable.

This isn’t a watertight argument, but it seems pretty clear that notation systems for math and logic are invented. In Part II, I’ll look at what it means to think of math as invented or discovered.

Nice post!
I do think that there’s something relevant to be said here for how we define concepts at all. For example, the concept of ‘rain’ was invented in the sense that we choose what to include with the concept of rain and what is not included. You could imagine a culture that defines “rain” as both the clouds and the liquid precipitation falling from those clouds, whereas our culture defines clouds and rain as separate/distinct concepts, even if they are causally related to one another. Where we choose to define the boundaries of concepts is dependent on human minds, and thus what is or isn’t defined as “rain” by this or that culture or individual is/was discovered by the first person to have coined that particular concept (even if that discoverer can never be known due to pragmatic constraints of obtaining such information likely to be forever lost in the past).
Although your point is well taken, i.e. that because “rain” is a ubiquitous, natural occurrence, it is rather odd to think of it as being discovered. But part of that intuition, I believe, relies on the fact that so many people have a common concept of “rain” across many cultures, and because the concept has been around for so long, it seems that it’s just always been there. And while the physical phenomena that we define as clouds and rain have always been there, even before the concepts were ever delineated for such phenomena, those concepts nevertheless depend on human minds, and thus there’s technically a first discoverer for any concept, in that they are the inventor of that concept. They’ve coined the term in a way that includes some set or conjunction of qualities and properties, while excluding others.
Anyway, I don’t think this necessarily detracts from anything you’ve said here, but I thought it was relevant to point out that concepts are invented (they didn’t use to exist and then one day they were fashioned by a person or culture), and thus we need to be careful to distinguish between the physical phenomena that ground a concept, and the abstraction itself which is representing that phenomena.
“The objects of discovery, on the other hand, are mind independent — they would have been there whether or not we were around to find them.”
I would disagree with this claim of yours, barring a certain qualification on what is meant by “object”. We discover the phenomena AS some set or conjunction of properties, based on how our brain’s pattern recognition schema infers the causal structure (even its being an “object” at all, among other things), and we further modify that concept’s conjunction of properties using higher level cognitive processes that operate within the constraints of language and the social need for communicability. Anyway, it’s all very interesting either way!

Hi Aaron,
Interesting…My conclusion is based on the following: If concepts require fracturing the world in some way, e.g. fracturing it into sets, categories, properties, conjunctions of properties, boundaries, etc., then whatever does the fracturing is required for concepts. My assumption is that the brain is what does the fracturing, based on its neurological schema, ergo, the brain is needed for concepts.
We’re all entitled to our opinion on the matter, but I see this as an inescapable conclusion. Are you saying that you are a Platonist of some kind, believing that concepts exist in some other realm, independent of minds/brains?